Q: Which of the following is the correct reciprocal identity for $$\sin \theta$$?

$$\sin \theta = \dfrac{1}{\csc \theta}$$

Q: Which trigonometric functions are always positive in quadrant II?

$$\sin \theta$$ and $$\csc \theta$$

Q: Given $$\tan \theta = \dfrac{y}{x}$$, which of the following is the correct quotient identity for $$\tan \theta$$?

$$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$$

Q: Which of the following is NOT a Pythagorean identity?

$$\tan^2$ \theta + \cot^2$ \theta = 1$$

Q: Which of the following is the correct reciprocal identity for $$\tan \theta$$?

$$\tan \theta = \dfrac{1}{\cot \theta}$$

Q: Which of the following is a correct way to derive the identity $$1 + \cot^2$ \theta = \csc^2$ \theta$$?

Divide $$\sin^2$ \theta + \cos^2$ \theta = 1 by$$ $$\sin^2$ \theta$$

Q: Which of the following is the correct quotient identity for $$\cot \theta$$?

$$\cot \theta = \dfrac{\cos \theta}{\sin \theta}$$

Question 1

Which of the following is the correct reciprocal identity for $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \dfrac{1}{\csc 	heta}$$

Question 2

If $$\cos 	heta = -\dfrac{3}{5}$$ and $$	heta is in $$quadrant III, what is $$\sin 	heta$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 3

Which trigonometric functions are always positive in quadrant II?

Accepted Answer

$$\sin 	heta$$ and $$\csc 	heta$$

Question 4

Given $$	an 	heta = \dfrac{y}{x}$$, which of the following is the correct quotient identity for $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \dfrac{\sin 	heta}{\cos 	heta}$$

Question 5

If $$\sin 	heta = 0.6$$ and $$	heta is in $$quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$-0.8$$

Question 6

Which of the following is NOT a Pythagorean identity?

Accepted Answer

$$	an^2$ 	heta + \cot^2$ 	heta = 1$$

Question 7

A ladder leans against a wall making an angle $$	heta$$ with the ground. If the top of the ladder is 12 ft above the ground and the base is 5 ft from the wall, what is $$\sin 	heta$$?

Accepted Answer

$$\dfrac{12}{13}$$

Question 8

If $$\sec 	heta = -2$$ and $$	heta is in $$quadrant III, what is $$	an 	heta$$?

Accepted Answer

$$\sqrt{3}$$

Question 9

Which of the following statements about the range of $$\sin 	heta is $$correct?

Accepted Answer

$$-1 \leq \sin 	heta \leq 1$$

Question 10

If $$	an 	heta = -3$$ and $$	heta is in $$quadrant IV, what is $$\sec 	heta$$?

Accepted Answer

$$\sqrt{10}$$

Question 11

Which of the following is the correct reciprocal identity for $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \dfrac{1}{\cot 	heta}$$

Question 12

A point $$P(x, y) is on $$the terminal side of angle $$	heta in $$standard position, and $$r = \sqrt{x^2$ + y^2$}. $$Which of the following is the correct definition for $$\sec 	heta$$?

Accepted Answer

$$\sec 	heta = \dfrac{r}{x}$$

Question 13

If $$\sin 	heta = -\dfrac{5}{13}$$ and $$	heta is in $$quadrant III, what is $$\cos 	heta$$?

Accepted Answer

$$-\dfrac{12}{13}$$

Question 14

Which of the following is a correct way to derive the identity $$1 + \cot^2$ 	heta = \csc^2$ 	heta$$?

Accepted Answer

Divide $$\sin^2$ 	heta + \cos^2$ 	heta = 1 by$$ $$\sin^2$ 	heta$$

Question 15

If $$\cos 	heta = 0.5$$ and $$	heta is in $$quadrant IV, what is $$\sin 	heta$$?

Accepted Answer

$$-\dfrac{\sqrt{3}}{2}$$

Question 16

Which of the following is the correct range for $$\sec 	heta$$?

Accepted Answer

$$|\sec 	heta| \geq 1$$

Question 17

A Ferris wheel has a radius of 20 meters. If a seat is at a height $$y$$ meters above the center, what is $$\sin 	heta$$ for the angle $$	heta$$ the seat makes with the horizontal?

Accepted Answer

$$\dfrac{y}{20}$$

Question 18

If $$	an 	heta = 2$$ and $$	heta is in $$quadrant III, what is $$\sin 	heta$$?

Accepted Answer

$$-\dfrac{2}{\sqrt{5}}$$

Question 19

Which of the following is the correct quotient identity for $$\cot 	heta$$?

Accepted Answer

$$\cot 	heta = \dfrac{\cos 	heta}{\sin 	heta}$$

Trigonometric Identities: Reciprocal, Pythagorean, and Quotient Identities

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